A Polynomial Chaos Approach to Control Design

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Abstract

A method utilizing H2 control concepts and the numerical method of Polynomial Chaos was developed in order to create a novel robust probabilistically optimal control approach. This method was created for the practical reason that uncertainty in parameters tends to be inherent in system models. As such, the development of new methods utilizing probability density functions (PDFs) was desired.
From a more theoretical viewpoint, the utilization of Polynomial Chaos for studying and designing control systems has not been very thoroughly investigated. The current work looks at expanding the H2 and related Linear Quadratic Regulator (LQR) control problems for systems with parametric uncertainty. This allows solving deterministic linear equations that represent probabilistic linear differential equations. The application of common LTI (Linear Time Invariant) tools to these expanded systems are theoretically justified and investigated. Examples demonstrating the utilized optimization process for minimizing the H2 norm and parallels to LQR design are presented.
The dissertation begins with a thorough background section that reviews necessary probability theory. Also, the connection between Polynomial Chaos and dynamic systems is explained. Next, an overview of related control methods, as well as an in-depth review of current Polynomial Chaos literature is given. Following, formal analysis, related to the use of Polynomial Chaos, is provided. This lays the ground for the general method of control design using Polynomial Chaos and H2. Then an experimental section is included that demonstrates controller synthesis for a constructed probabilistic system. The experimental results lend support to the method.